Polynomial identities from weighted lattice path counting

We give bijective proofs, using weighted lattice paths, of two multinomial identities concerning the generalized h-factorial polynomials of order n.

x]ⁿ_h:=x(x + h)(x +2h)(x + (n − 1)h)

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The first-one is the multinomial identity of order s verified by these polynomials. Using this identity (and its proof) as a lemma, we derive the main identity that generalizes previous results of Carlitz (1977), Egorychev (1974) and the classical identity of Banach.